Variational Quantum Continuous Optimization: a Cornerstone of Quantum Mathematical Analysis

Abstract: Here we show how universal quantum computers based on the quantum circuit model can handle mathematical analysis calculations for functions with continuous domains, without any digitalization, and with remarkably few qubits. The basic building block of our approach is a variational quantum circuit where each qubit encodes up to three continuous variables (two angles and one radious in the Bloch sphere). By combining this encoding with quantum state tomography, a variational quantum circuit of $n$ qubits can optimize functions of up to $3n$ continuous variables in an analog way. We then explain how this quantum algorithm for continuous optimization is at the basis of a whole toolbox for mathematical analysis on quantum computers. For instance, we show how to use it to compute arbitrary series expansions such as, e.g., Fourier (harmonic) decompositions. In turn, Fourier analysis allows us to implement essentially any task related to function calculus, including the evaluation of multidimensional definite integrals, solving (systems of) differential equations, and more. To prove the validity of our approach, we provide benchmarking calculations for many of these use-cases implemented on a quantum computer simulator. The advantages with respect to classical algorithms for mathematical analysis, as well as perspectives and possible extensions, are also discussed.